Results 1 to 6 of 6
Like Tree1Thanks
  • 1 Post By richard1234

Math Help - Trying to work through a Fibonacci problem using Binet's Formula

  1. #1
    Newbie
    Joined
    Aug 2012
    From
    Salem, Oregon
    Posts
    10

    Trying to work through a Fibonacci problem using Binet's Formula

    I am trying to show that the square of any Fibonacci Number differs by one from the product of its neighbors:


     (F_n)^2 - ((F_{n-1}) \times (F_{n+1})) = \pm 1


    My best attempt is to substitute Binet's Formula:


     \left( \frac { ( 1 + \sqrt{5} )^n - (1 - \sqrt{5} )^n } { 2^n \sqrt{5} } \right)^2 - \left( \frac { ( 1 + \sqrt{5} )^{n-1} - (1 - \sqrt{5} )^{n-1} } { 2^{n-1} \sqrt{5} } \times \frac { ( 1 + \sqrt{5} )^{n+1} - (1 - \sqrt{5} )^{n+1} } { 2^{n+1} \sqrt{5} } \right)


    and simplify, but keep getting buggered near the end. I get down to:


     \frac { ( ( 1 + \sqrt{5} )^{n-1} ( 1 - \sqrt{5} )^{n+1} ) + ( ( 1 - \sqrt{5} )^{n-1} ( 1 + \sqrt{5} )^{n+1} ) - 2 ( -4 )^n } { 5 ( 4^n ) }


    and can go no further due to the unequal bases. I know (because even values of "n" yield -1) that:


     { ( ( 1 + \sqrt{5} )^{n-1} ( 1 - \sqrt{5} )^{n+1} ) + ( ( 1 - \sqrt{5} )^{n-1} ( 1 + \sqrt{5} )^{n+1} ) = - 3 ( -4 )^n }


    which would make the final formula:


     \frac { - 5 ( -4 )^n } { 5 ( 4^n ) } = \pm 1


    but I seem to be missing something. I could swear I've seen this done, but can't find it anywhere. Any pointers as to what I'm missing?


    - Stephen
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Aug 2012
    From
    Salem, Oregon
    Posts
    10

    Re: Trying to work through a Fibonacci problem using Binet's Formula

    I realized this morning that I had overlooked some of my algebra:

    x^{n-1} = \frac{x^n}{x} and x^{n+1} = x^n \times x

    So, moving through a few more steps I get to:

    \frac{ \left( \frac{{(-4)^n-(-4)^n \sqrt{5}}}{(1+ \sqrt{5})} \right)+\left( \frac{{(-4)^n+(-4)^n \sqrt{5}}}{(1- \sqrt{5})} \right)-2(-4)^n}{5(4^n)}

    I'm not sure if it's any better, but it's something ...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Trying to work through a Fibonacci problem using Binet's Formula

    Wikipedia has a nice proof of this theorem (Cassini's identity):

    Cassini and Catalan identities - Wikipedia, the free encyclopedia
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Aug 2012
    From
    Salem, Oregon
    Posts
    10

    Re: Trying to work through a Fibonacci problem using Binet's Formula

    That is nice - I was hoping for something algebraic, but this is about as straightforward as I could ask for. I have two concerns with this solution though:

    1. Using a matrix looks an awful lot like we simply restated the problem and called it solved
    2. While creating the matrix using F_n with an odd value of n works well, won't using an even value of n yield a determinant of 1^n which will always be positive?

    It looks like I'll have to brush up on matrices so I can explain it. Wikipedia to the rescue once again!

    Thank you!

    - Stephen
    Last edited by stephyounger; August 7th 2012 at 07:45 AM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jun 2012
    From
    AZ
    Posts
    616
    Thanks
    97

    Re: Trying to work through a Fibonacci problem using Binet's Formula

    1. But you are definitely allowed to write F_{n-1}F_{n+1} - F_{n}^2 as the determinant of a 2x2 matrix, right?

    2. Yes, but 1^n = 1 for even n.
    Thanks from stephyounger
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Aug 2012
    From
    Salem, Oregon
    Posts
    10

    Re: Trying to work through a Fibonacci problem using Binet's Formula

    Yes on both counts. My "concerns" were rather poorly worded - what I meant to convey is that I'm worried my unfamiliarity with matrices leaves me unprepared to answer what should be a couple of simple questions, not that there was a problem with the proof. Hence my need to "brush up."

    - Stephen
    Last edited by stephyounger; August 7th 2012 at 09:00 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 5
    Last Post: May 7th 2012, 08:51 PM
  2. formula remains true Fibonacci sequence
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: December 4th 2011, 02:47 PM
  3. Help with work problem/formula
    Posted in the Algebra Forum
    Replies: 4
    Last Post: February 3rd 2011, 07:48 AM
  4. Binet's formula
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: September 28th 2010, 01:13 PM
  5. Binet's formula
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: September 15th 2010, 03:05 PM

Search Tags


/mathhelpforum @mathhelpforum